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%I A051716 #39 Apr 23 2023 07:28:38
%S A051716 1,-1,-1,-1,-1,1,1,-1,-1,1,5,-5,-691,691,7,-7,-3617,3617,43867,-43867,
%T A051716 -174611,174611,854513,-854513,-236364091,236364091,8553103,-8553103,
%U A051716 -23749461029,23749461029,8615841276005,-8615841276005,-7709321041217,7709321041217,2577687858367
%N A051716 Numerators of Bernoulli twin numbers C(n).
%C A051716 The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to _Paul Curtz_.
%C A051716 For denominators see A051717.
%C A051716 Negatives of numerators of column 1 of table described in A051714/A051715.
%H A051716 Vincenzo Librandi, <a href="/A051716/b051716.txt">Table of n, a(n) for n = 0..640</a>
%H A051716 M. Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.
%F A051716 Numerators of differences of the sequence of rational numbers 0 followed by A164555/A027642. - _Paul Curtz_, Jan 29 2017
%F A051716 The e.g.f. of the rationals a(n)/A051717(n) is -(1/x + x^2/2 + x/(1 - exp(x)) + dilog(exp(-x))), (with dilog(x) = polylog(2, 1-x)). From integrating the e.g.f. of the z-sequence  (exp(x) - (1+x))/(exp(x) -1)^2 for the Bernoulli polynomials of the second kind (A290317 / A290318). - _Wolfdieter Lang_, Aug 07 2017
%e A051716 The C(n) sequence is 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...
%p A051716 C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;
%t A051716 c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Numerator[c[n]], {n,0,34}] (* _Jean-François Alcover_, Dec 19 2011 *)
%o A051716 (PARI) a(n) = if (n==0, 1, nu = numerator(bernfrac(n)+bernfrac(n-1)); if (n%2, -nu, nu)); \\ _Michel Marcus_, Jan 29 2017
%o A051716 (Magma)
%o A051716 f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
%o A051716 function A051716(n)
%o A051716   if n eq 0 then return 1;
%o A051716   elif (n mod 2) eq 0 then return Numerator(f(n));
%o A051716   else return Numerator(-f(n));
%o A051716   end if;
%o A051716 end function;
%o A051716 [A051716(n): n in [0..50]]; // _G. C. Greubel_, Apr 22 2023
%o A051716 (SageMath)
%o A051716 def f(n): return bernoulli(n)+bernoulli(n-1)
%o A051716 def A051716(n):
%o A051716     if (n==0): return 1
%o A051716     elif (n%2==0): return numerator(f(n))
%o A051716     else: return numerator(-f(n))
%o A051716 [A051716(n) for n in range(51)] # _G. C. Greubel_, Apr 22 2023
%Y A051716 Cf. A000367, A027641, A027642, A051714, A051715.
%Y A051716 Cf. A051717, A129825, A129826, A129724, A164555.
%K A051716 sign,easy,nice,frac
%O A051716 0,11
%A A051716 _N. J. A. Sloane_
%E A051716 More terms from _James A. Sellers_, Dec 08 1999
%E A051716 Edited by _N. J. A. Sloane_, May 25 2008

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